I was asked this question a few minutes ago on Twitter. I have heard that question brought up before.
This question seems to matter as some sort of a legal issue.
Here is a definition of mental illness from the Mayo Clinic
Mental illness refers to a wide range of mental health conditions – dosorders that affect your mood, thinking and behavior. Examples of mental illness include depression, anxiety disorders, schizophrenia, eating disorders and addictive behaviors.
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While it may be splitting hairs to accept one term over the other, the medical definition of a mental disorder as a physical or mental condition that is not normal or healthy seems to fit. However unlike many mental disorders, there are discernible neurological impairments that cause dementia.